Mumbai University > Information Technology, Computer Engineering > Sem 3 > Applied Mathematics 3

Marks: 6M

Year: May 2014

we have $w = \frac{2}{z+i}$

$z + i = \frac{2}{w}$

Where z = x + i y and w = u +i v

$x+ i y +i = \frac{2}{u+iv} × \frac{u-iv}{u-iv}$ (conjugate)

Therefore $x + i(y+ 1) = \frac{2(u-iv)}{u^2+v^2}$

Therefore $x +i(y+1) = \frac{2u-2iv}{u^2+v^2}$

Equating of real and imaginary parts

$X = \frac{2u}{u^2 + v^2} , y+1 = \frac{-2v}{u^2 + v^2}$

Equation of real axis on z -plane is y = 0

Therefore $0+1 = \frac{-2v}{u^2+v^2}$

Therefore $u^2 + v^2 = -2v$

Therefore $u^2 + v^2 + 2v = 0$ which is circle in w plane

Comparing this equation with equation of circle

$u^2 + v^2 + 2gu + 2fv + c = 0$

Therefore g=0, f= 1,c=0

Center (-g,-f) = (0,-1) and radius = $\sqrt{g^2 + f^2 - c} = 1$

Therefore real axis of z -plane is mapped to circle in w – plane.